Why is my cloth material not rendering properly?
Sorry for the misunderstanding - you meant that the flag does not move. I thought you meant the image. It is good habit to cache all physics simulation before rendering, so the renderer has all the data needed to render the animation. Go to the Physics pane with the flag selected check 'Disk Cache' and after that hit 'Bake' in the Cloth Cache tab. After completing the caching process you can render and everything works just fine. It should look similar to the image below before rendering